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In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity problems are part of geometric measure theory.

History

Various specialized forms of the problem were solved, but it was only in 1930 that general solutions were found in the context of mappings (immersions) independently by Jesse Douglas and Tibor Radó. Their methods were quite different; Radó's work built on the previous work of René Garnier and held only for rectifiable simple closed curves, whereas Douglas used completely new ideas with his result holding for an arbitrary simple closed curve. Both relied on setting up minimization problems; Douglas minimized the now-named Douglas integral while Radó minimized the "energy". Douglas went on to be awarded the Fields Medal in 1936 for his efforts.

In higher dimensions

The extension of the problem to higher dimensions (that is, for $k$ -dimensional surfaces in $n$ -dimensional space) turns out to be much more difficult to study. Moreover, while the solutions to the original problem are always regular, it turns out that the solutions to the extended problem may have singularities if $k\leq n-2$ . In the hypersurface case where $k=n-1$ , singularities occur only for $n\geq 8$ . An example of such singular solution of the Plateau problem is the Simons cone, a cone over $S^{3}\times S^{3}$ in $\mathbb {R} ^{8}$ that was first described by Jim Simons and was shown to be an area minimizer by Bombieri, De Giorgi and Giusti.^[1] To solve the extended problem in certain special cases, the theory of perimeters (De Giorgi) for codimension 1 and the theory of rectifiable currents (Federer and Fleming) for higher codimension have been developed. The theory guarantees existence of codimension 1 solutions that are smooth away from a closed set of Hausdorff dimension $n-8$ . In the case of higher codimension Almgren proved existence of solutions with singular set of dimension at most $k-2$ in his regularity theorem. S. X. Chang, a student of Almgren, built upon Almgren’s work to show that the singularities of 2-dimensional area minimizing integral currents (in arbitrary codimension) form a finite discrete set.^[2]^[3]

The axiomatic approach of Jenny Harrison and Harrison Pugh^[4] treats a wide variety of special cases. In particular, they solve the anisotropic Plateau problem in arbitrary dimension and codimension for any collection of rectifiable sets satisfying a combination of general homological, cohomological or homotopical spanning conditions. A different proof of Harrison-Pugh's results were obtained by Camillo De Lellis, Francesco Ghiraldin and Francesco Maggi.^[5]

Physical applications

Physical soap films are more accurately modeled by the $(M,0,\Delta )$ -minimal sets of Frederick Almgren, but the lack of a compactness theorem makes it difficult to prove the existence of an area minimizer. In this context, a persistent open question has been the existence of a least-area soap film. Ernst Robert Reifenberg solved such a "universal Plateau's problem" for boundaries which are homeomorphic to single embedded spheres.

References

^ Bombieri, Enrico; De Giorgi, Ennio; Giusti, Enrico (1969), "Minimal cones and the Bernstein problem", Inventiones Mathematicae, 7 (3): 243–268, Bibcode:1969InMat...7..243B, doi:10.1007/BF01404309, S2CID 59816096
^ Chang, Sheldon Xu-Dong (1988), "Two-dimensional area minimizing integral currents are classical minimal surfaces", Journal of the American Mathematical Society, 1 (4): 699–778, doi:10.2307/1990991, JSTOR 1990991
^ De Lellis, Camillo (2016), "Two-dimensional almost area minimizing currents" (PDF), Bollettino dell'Unione Matematica Italiana, 9 (1): 3–67, doi:10.1007/s40574-016-0057-1, MR 3470822
^ Harrison, Jenny; Pugh, Harrison (2017), "General Methods of Elliptic Minimization", Calculus of Variations and Partial Differential Equations, 56 (1), arXiv:1603.04492, doi:10.1007/s00526-017-1217-6, S2CID 119704344
^ De Lellis, Camillo; Ghiraldin, Francesco; Maggi, Francesco (2017), "A direct approach to Plateau's problem" (PDF), Journal of the European Mathematical Society, 19 (8): 2219–2240, doi:10.4171/JEMS/716, S2CID 29820759

Douglas, Jesse (1931). "Solution of the problem of Plateau". Trans. Amer. Math. Soc. 33 (1): 263–321. doi:10.2307/1989472. JSTOR 1989472.
Reifenberg, Ernst Robert (1960). "Solution of the {Plateau} problem for m-dimensional surfaces of varying topological type". Acta Mathematica. 104 (2): 1–92. doi:10.1007/bf02547186.
Fomenko, A.T. (1989). The Plateau Problem: Historical Survey. Williston, VT: Gordon & Breach. ISBN 978-2-88124-700-2.
Morgan, Frank (2009). Geometric Measure Theory: a Beginner's Guide. Academic Press. ISBN 978-0-12-374444-9.
O'Neil, T.C. (2001) [1994], "Geometric Measure Theory", Encyclopedia of Mathematics, EMS Press
Radó, Tibor (1930). "On Plateau's problem". Ann. of Math. 2. 31 (3): 457–469. Bibcode:1930AnMat..31..457R. doi:10.2307/1968237. JSTOR 1968237.
Struwe, Michael (1989). Plateau's Problem and the Calculus of Variations. Princeton, NJ: Princeton University Press. ISBN 978-0-691-08510-4.
Almgren, Frederick (1966). Plateau's problem, an invitation to varifold geometry. New York-Amsterdam: Benjamin. ISBN 978-0-821-82747-5.

Harrison, Jenny; Pugh, Harrison (2016). Open Problems in Mathematics (Plateau's Problem). Springer. arXiv:1506.05408. doi:10.1007/978-3-319-32162-2. ISBN 978-3-319-32160-8.

This article incorporates material from Plateau's Problem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.